Near-infinity concentrated norms and the fixed point property for nonexpansive maps on closed, bounded, convex sets

NEAR-INFINITY CONCENTRATED NORMS AND THE FIXED POINT PROPERTY FOR NONEXPANSIVE MAPS ON CLOSED, BOUNDED, CONVEX SETS F.E. CASTILLO-S´ ANTOS, P.N. DOWLING, H. FETTER, M. JAP ´ ON, C.J. LENNARD, B. SIMS, B. TURETT Abstract. In this paper we define the concept of a near-infinity concentrated norm on a Banach space Xwith a boundedly complete Schauder basis. When k·k is such a norm, we prove that (X, k·k) has the fixed point property (FPP); that is, every nonexpansive self-mapping defined on a closed, bounded, convex subset has a fixed point. In particular, P.K. Lin’s norm in `1[14] and the norm νp(·) (with p= (pn) and limnpn= 1) introduced in [3] are examples of near-infinity concentrated norms. When νp(·) is equivalent to the `1-norm, it was an open problem as to whether (`1, νp(·)) had the FPP. We prove that the norm νp(·) always generates a nonreflexive Banach space X=R⊕p1(R⊕p2(R⊕p3...)) satisfying the FPP, regardless of whether νp(·) is equivalent to the `1-norm. We also obtain some stability results. 1. Introduction and Preliminaries Let (X, k · k) be a Banach space and Ca subset of X. A mapping T:C→Cis said to be nonexpansive if kT x −Tyk ≤ kx−ykfor every x, y ∈C. The Banach space Xendowed with the norm k·khas the fixed point property (FPP) if every nonexpansive mapping defined from a closed bounded convex subset Cof Xinto itself has a fixed point. This property is not preserved by isomorphism, that is, it strongly depends on the underlying norm [14] . There is a wide literature relating geometric properties of reflexive Banach spaces with the fulfilment of the fixed point property (see, for instance, the monographs [9], [13] and the references therein). The Banach space `1endowed with its standard norm k·k1is a classical example of a nonreflexive Banach space that fails to have the FPP. It is possible to “perturb” this (`1,k·k1)-example to obtain other Banach spaces that fail to have the FPP. One such class of Banach spaces are those that contain asymptotically isometric copies of `1. Recall that a Banach space (X, k · k) contains an asymptotically isometric copy (a.i.c.) of `1if there are a sequence (xn)⊂Xand a decreasing sequence (n)⊂(0,1) with limnn= 0 such that ∞ X n=1 (1 −n)|tn| ≤      ∞ X n=1 tnxn     ≤ ∞ X n=1 |tn| 1991 Mathematics Subject Classification. 46B03, 47H09, 47H10. Key words and phrases. fixed point property; nonexpansive mappings; renorming theory. The third author is partially funded by CONACYT grants 243722 and 613207. The fourth author is partially supported by MCIN, Grant MTM2015-65242-C2-1-P and Andalusian Regional Government Grant FQM-127. She is also grateful to the Department of Mathematics of the University of Pittsburgh for its support while preparing this manuscript. The fifth author thanks the University of Newcastle for its financial support during part of the preparation of this paper; and Brailey, Gail and the Mathematics Department for their hospitality. 1 2 CASTILLO-S´ ANTOS, DOWLING, FETTER, JAP ´ ON, LENNARD, SIMS, TURETT for every (tn)∈`1. It was proved in [4] that if a Banach space contains an a.i.c. of `1then it fails to have the FPP. It turns out that there exist equivalent norms on `1which fail to contain an a.i.c. of `1. Let us state some examples: •The so-called P.K. Lin norm, defined as |||x|||L:= sup k≥1 γk ∞ X n=k |x(n)|;x= ∞ X n=1 x(n)en where (γk) is a nondecreasing sequence in (0,1) with limkγk= 1. In [5] it was proved that (`1,||| · |||L) fails to contain an a.i.c. of `1. Later on, P.K. Lin [14] proved that (`1,|||·|||L) has the FPP for γk:= 8k 1+8k. This condition was extended to every sequence (γk) with limkγk= 1 (see [7] and [11]). P. K. Lin’s result opened new avenues of research in the fixed point theory of nonexpansive mappings, since he settled negatively the long-standing open question: “Does the fixed point property imply reflexivity?” Since then, many other articles have appeared obtaining sufficient conditions that imply the FPP for equivalent norms on `1(see for instance [2, 6, 7, 8, 10, 11, 12, 15]). •Fix a nonincreasing sequence p= (pn)n⊂(1,+∞) with limnpn= 1. In the sequence space c00 of all real sequences with finitely many non-null coordinates, we define the norm νp(x) = limnνn(p, x) where ν1(p, x) := |x1|, νn+1(p, x) := (|x1|p1+νn(Sp, Sx)p1)1/p1, with x= (x1, x2, ...) and Sz := (z2, z3, ...) when z= (z1, z2, ...). The completion of c00 with the νp(·) norm gives us a Banach space Xwith aboundedly complete Schauder basis (en). Also, Xis the set of all real sequences x= (xn) for which νp(x) := supnνn(p, x) = limnνn(p, x)<∞; which we summarize by writing X=R⊕p1(R⊕p2(R⊕p3. . . )). Let q= (qn) be the sequence satisfying 1 pn+1 qn= 1 for every n∈N. Whenever the sequence (pn) converges to 1 quickly enough, the norm νp(·) provides an equivalent norm in `1; that is, (X, νp(·)) and `1are isomorphic Banach spaces. In fact, it was proved in [3, Proposition 1] that νp(·) is equivalent to the `1norm if and only if there exists some δ > 0 so that qn≥δlog nfor all n∈N. It is also known that (`1, νp(·)) fails to contain asymptotically isometric copies of `1[3, Theorem 1]. However, unlike P.K. Lin’s norm, it was unknown whether `1with the norm νp(·) had the fixed point property. In what follows, we enlarge the class of norms on `1satisfying the FPP and we include, as a particular case, the norm νp(·) defined in [3]. We will extend our result to a more general framework. For instance, we will prove the fulfilment of the FPP for (X, νp(·)) even when this norm fails to be an `1-norm. Furthermore, we obtain stability of the fixed point property for certain norms along rays emanating from near-infinity concentrated norms. 2. Near-infinity concentrated norms and the FPP Throughout this paper, let Xdenote a Banach space with a Schauder basis {en}n. Given x= ∞ X n=1 x(n)en∈X, we denote by supp(x) = {n∈N:x(n)6= 0}, Qk(x) = ∞ X n=k x(n)enand Pk(x) = k−1 X n=1 x(n)en(P1= 0). The basis is said to be premonotone for the norm k·kwhen kQkk ≤ 1 for every k∈N. NEAR-INFINITY CONCENTRATED NORMS AND THE FPP 3 Given k∈Nand x∈X, we write k≤xwhenever k≤min{supp(x)}and k < x whenever k < min{supp(x)}. We say that (yn) is a block basic sequence for {en}n if it is bounded and there exist positive integers p1≤q1< p2≤q2< ... such that ynbelongs to the span of {epn,· · · , eqn}for every n∈N. The Schauder basis is said to be boundedly complete if sup n     n X i=1 tiei     <+∞implies that ∞ X i=1 tiei∈X. When the Schauder basis (en) is boundedly complete, the Banach space Xis isomorphic to a dual space Z∗, where Zis the closed subspace spanned by the biorthogonal functionals (e∗ n) in X∗. In this case, we can consider in Xthe weak∗topology σ(X, Z), for which the convergence coincides with the coordinate-to-coordinate convergence for norm-bounded sequences. Moreover, the closed unit ball is σ(X, Z)-sequentially compact and therefore every bounded sequence in Xhas a subsequence which converges coordinatewise (see for instance Theorem 3.2.10 in [1]). In what follows the weak∗topology always refers to the σ(X, Z) topology for Banach spaces with boundedly complete Schauder basis. In the case where X=`1endowed with the standard Schauder basis, this w∗-topology coincides with the σ(`1, c0) topology. Definition 2.1. [2] A norm ||| · ||| on a Banach space Xwith a Schauder basis {en}is said to be a sequentially separating norm if for every  > 0there exists some k∈Nsuch that |||x||| + lim sup n |||xn||| ≤ (1 + ) lim sup n |||x+xn||| whenever k≤xand (xn)nis a block basic sequence of {en}nin X. Definition 2.2. Let Xbe a Banach space with a Schauder basis {en}nand let ||| · ||| be a norm on X. This norm is called near-infinity concentrated (n.i.c.) if it has the following properties: (1) It is a sequentially separating norm. (2) It is premonotone. (3) There exist R0>5and M∈[0,1) such that for every k∈N, there exists a function Fk: (0,+∞)→[0,+∞)satisfying the following conditions: (a) limλ→0+ Fk(λ) λ≤M R0 . (b) For every bounded pointwise-null sequence (xn)with lim infn|||xn||| ≥ 1, for all λ∈(0,+∞), and for every z∈Xwith Qk(z)=0and |||z||| ≤ R0, lim sup n |||xn+λz||| ≤ lim sup n |||xn||| +Fk(λ)|||z||| . Remark 2.3. Observe that Property (3) can be re-written as: There exists K≥0 such that for every k∈N, there exists a function Fk: (0,+∞)→[0,+∞)satisfying (a)’ and (b); where condition (a)’ is: limλ→0+ Fk(λ) λ≤K < 1 5. Given K, we may take M:= 1 −1−5K K+1 =6K K+1 and R0:= 5 + 1−5K K+1 =6 K+1 . Note that if ||| · ||| is an equivalent norm on `1satisfying akkQk(x)k1≤ |||Qk(x)||| ≤ bkkQk(x)k1,for all x∈`1, 4 CASTILLO-S´ ANTOS, DOWLING, FETTER, JAP ´ ON, LENNARD, SIMS, TURETT for every k∈N, with 0 < ak≤bkand limkbk/ak= 1, then it is clear that ||| · ||| is a sequentially separating norm. Nevertheless, there exist some equivalent norms on `1which do not satisify this condition but they are still sequentially separating [2, Example 3.2]. Furthermore, there exist Banach spaces with sequentially separating norms that are not isomorphic to `1, although the existence of such a norm implies that the Banach space Xis “similar” to `1, in the sense that it has the Schur property, and so is hereditarily `1[2, Corollary 7.4]. Recall that a Banach space X is hereditarily `1if each infinite dimensional closed subspace of Xcontains a further subspace isomorphic to `1. This implies, in particular, that if a Banach space with an unconditional Schauder basis has a sequentially separating norm, then the basis is boundedly complete, since otherwise Xwould contain an isomorphic copy of c0 (see for instance [1, Theorem 3.3.2]). Also note that in Definition 2.2, Property (3)(b), if (xn) is an arbitrary sequence of “bump functions sliding towards infinity”, each with their ||| · |||-norm asymptotically no less than 1, then lim supn|||xn+λz||| − lim supn|||xn||| λ is smaller than one would expect from just the triangle inequality: for all z=Pk(z) with |||z||| ≤ R0, for all λpositive and very small, the “upper asymptotic value” of the norm of xnis changed less than expected when we perturb each, xnby λz, since Fk(λ)R0/λ is approximately bounded by M < 1. In this sense, |||·||| is “nearinfinity concentrated”. Moreover, this third property prevents Xfrom containing an asymptotically isometric copy of `1, which we will now prove. Lemma 2.4. Let Xbe a Banach space with a boundedly complete Schauder basis. If ||| · ||| is an equivalent norm in Xsatisfying property (3) in Definition 2.2, then (X, ||| · |||)fails to have an a.i.c. of `1. Proof. Assume to the contrary that there exists a basic sequence (xn) in Xgenerating an a.i.c. of `1, that is, there is a decreasing sequence (n)⊂(0,1) with limnn= 0 such that ∞ X n=1 (1 −n)|tn| ≤  ∞ X n=1 tnxn ≤ ∞ X n=1 |tn|. By extracting a subsequence, we can assume that (xn) is w∗-convergent and, by replacing (xn) by ((x2n−x2n−1)/2)), that it is w∗-convergent to the null vector. Finally, using the sliding hump method and the fact that asymptotically isometric copies are stable by adding norm-null sequences, we can assume that the sequence (xn) generating the a.i.c. of `1is a disjointly supported w∗-null sequence. Take R0>5 and M∈[0,1) as in (3) of Definition 2.2. By omitting the first few terms of the sequence (xn)n, we can also assume that 1<(R0−M)/R0. From the previous inequalities |||xn||| ≤ 1 for every n∈Nand limn|||xn||| = 1. Let k:= 1 + max{supp(x1)}. Since ||| · ||| satisfies property (3) of a near-infinity concentrated norm, there exists a function Fk(λ) such that limλ→0+Fk(λ)/λ ≤M R0, and for every λ > 0 lim sup n |||xn+λR0x1||| ≤ lim sup n |||xn||| +Fk(λ)R0|||x1||| ≤ 1 + Fk(λ)R0. On the other hand, for every n≥2, 1−n+λR0(1 −1)≤ |||xn+λR0x1|||. Letting ntend to infinity, we see that 1 + λR0(1 −1)≤1 + Fk(λ)R0 NEAR-INFINITY CONCENTRATED NORMS AND THE FPP 5 and so λ(1 −1)≤Fk(λ) for every λ > 0. Letting λ→0, we get that (1 − 1)≤limλ→0+Fk(λ) λ≤M R0, which implies that R0(1 −1)≤M, and this is a contradiction.  Before stating our main result, we recall some standard arguments used to prove the FPP (see for instance [14] or [11]): Let Cbe a closed bounded convex subset of a Banach space (X, k·k) and T:C→Cbe a nonexpansive mapping. Using Banach’s Contraction Mapping Theorem, we can always find a sequence (xn)⊂Csuch that limnkxn−Txnk= 0. Such sequences are called approximate fixed point sequences (a.f.p.s.). In fact, if (xn) is an a.f.p.s. and r > 0, the set {x∈C: lim sup n kxn−xk ≤ r} is either empty, or a non-empty closed convex T-invariant subset of C, in which we can find new approximate fixed point sequences. In a dual Banach space Xwith separable predual Y, every a.f.p.s. has a subsequence which is w∗-convergent. For example, if Xis a Banach space with a boundedly complete Schauder basis {en}n, and corresponding biorthogonal functionals {fn}n⊂X∗, then for Y:= the closed linear span of {fn}nin X∗,Y∗is isomorphic to Xand every a.f.p.s. in Xhas a subsequence that is σ(X, Y )-convergent. Therefore, we will subsequently assume that approximate fixed point sequences in bounded subsets of Xare w∗-convergent. Using Cantor’s theorem (see [14] or [11, Lemma 1]), the above argument lets us deduce that if T:C→Cis a fixed point free nonexpansive mapping, there exist some a > 0 and a closed convex T-invariant subset, denoted again by C, such that lim supnkyn−yk> a whenever (yn)⊂Cis an a.f.p.s. and y=w∗-lim yn. Note that from Definition 2.1 it is not difficult to check the following [2]: Lemma 2.5. A norm ||| · ||| in a Banach space Xwith a Schauder basis {en}nis sequentially separating if and only if limkSk(X, ||| · |||)=1, with Sk(X, ||| · |||) := sup |||x||| + lim supn|||xn||| lim supn|||x+xn||| , where the supremum is taken over all vectors x∈Xwith k≤xand all block basic sequences of {en}n. If we fix the norm ||| · ||| in the Banach space X, we will use Skto denote Sk(X, ||| · |||). Lemma 2.6. Let (X, |||·|||)be a Banach space with a boundedly complete Schauder basis {en}nsuch that ||| · ||| is premonotone and sequentially separating. The following holds: if (xn),(yn)are two sequences in Xthat are w∗-convergent to xand yrespectively, then lim sup m lim sup n |||xn−ym||| ≥ lim sup n |||xn−x||| + lim sup m |||ym−y|||. Proof. Let k∈Nand δ > 0 be given. Choose a subsequence (xn`) of (xn) such that lim sup n |||Qk(xn−x)||| = lim `|||Qk(xn`−x)||| . Fix m∈N. Then lim sup n |||xn−ym||| ≥ lim sup n |||Qk(xn−ym)||| = lim sup n |||Qk(xn−x)−Qk(ym−x)||| ≥lim sup ` |||Qk(xn`−x)−Qk(ym−x)||| . 6 CASTILLO-S´ ANTOS, DOWLING, FETTER, JAP ´ ON, LENNARD, SIMS, TURETT By the Bessaga-Pe lczynski Selection Principle (see, for example, [1, p. 14]), passing to a further subsequence if necessary, we may choose a block basic sequence (u`) of (en) such that Qk(u`) = u`and |||u`−Qk(xn`−x)||| < δ. Then lim sup n |||xn−ym||| ≥ lim sup ` |||u`−Qk(ym−x)||| − δ ≥1 Sk|||Qk(ym−x)||| + lim sup ` |||u`|||−δ ≥1 Sk|||Qk(ym−x)||| + lim `|||Qk(xn`−x)|||−2δ =1 Sk|||Qk(ym−x)||| + lim sup n |||Qk(xn−x)|||−2δ . Letting mtend to ∞; noting that Sk≥1; and using a perturbation argument similar to the one above then yields: lim sup m lim sup n |||xn−ym||| ≥ 1 Sklim sup m |||Qk(ym−x)||| + lim sup n |||Qk(xn−x)|||−2δ =1 Sklim sup n |||xn−x||| + lim sup m |||Qk(ym−y) + Qk(y−x)|||−2δ ≥1 Sk lim sup n |||xn−x||| +1 S2 k|||Qk(y−x)||| + lim sup m |||Qk(ym−y)|||−4δ ≥1 Sk lim sup n |||xn−x||| +1 S2 k lim sup m |||Qk(ym−y)||| − 4δ =1 Sk lim sup n |||xn−x||| +1 S2 k lim sup m |||ym−y||| − 4δ . In the above calculation, we used the fact that lim supn|||Qk(xn−x)||| = lim supn|||xn− x||| and lim supm|||Qk(ym−y)||| = lim supm|||ym−y||| for every k∈N. Since the above inequalities hold for every k∈N, letting ktend to infinity gives lim sup m lim sup n |||xn−ym||| ≥ lim sup n |||xn−x||| + lim sup m |||ym−y||| − 4δ , for every δ > 0. Since δ > 0 is arbitrary, we obtain the desired inequality.  Theorem 2.7. Let Xbe a Banach space with a boundedly complete Schauder basis and let ||| · ||| be a near-infinity concentrated (n.i.c.) norm on X. Then (X, ||| · |||) has the FPP, that is, every nonexpansive self-map on a closed bounded convex subset of Xhas a fixed point. Proof. Assume, to the contrary, that there exists a closed bounded convex subset Cof Xand T:C→Ca nonexpansive mapping such that b= inf{lim sup n |||yn−y||| : (yn)⊂Cis an a.f.p.s. and yn w∗ →y}>0. Without loss of generality we can assume that b= 1. We proceed as follows. Fix some 0 < 1<min{1 4(1 −M+1 2),1 10 (R0−5)}, where M∈[0,1) and R0>5 are the constants given by condition (3) in Definition 2.2. Consider an a.f.p.s. (xn) in Csuch that xn w∗ →x0∈Xand lim supn|||xn−x0||| < 1 + 1. Again, without loss of generality, we can assume that x0= 0 so that lim supn|||xn||| <1 + 1. Define the set D:= z∈C: lim sup n |||xn−z||| ≤ 2(1 + 1). NEAR-INFINITY CONCENTRATED NORMS AND THE FPP 7 Then Dis a closed convex T-invariant subset of C. Moreover, using the triangle inequality, lim sup m lim sup n |||xn−xm||| ≤ 2 lim sup n |||xn||| <2(1 + 1) ; so Dis not empty and we can assume that xn∈Dfor nlarge enough. Define c:= inf lim sup n |||yn−y||| : (yn)⊂Dis an a.f.p.s. and yn w∗ →y. Notice that 1 ≤c. To simplify the notation, we define A∗(D) := ny∈X:∃(yn)⊂Dan a.f.p.s. such that w∗- lim nyn=yo. We now prove that sup y∈A∗(D) |||y||| ≤ 4+41: Indeed, let (yn)⊂Dbe an a.f.p.s. with w∗-lim yn=y. In particular, lim supn|||xn− ym||| ≤ 2(1 + 1) for every m∈N. Using the triangle inequality and Lemma 2.6 (with x= 0): |||y||| ≤ lim sup m |||y−ym|||+lim sup n |||xn|||+lim sup m lim sup n |||xn−ym||| ≤ 4(1+1). Next we show that sup y∈A∗(D) |||Qk(y)||| ≤ µk:= 2S2 k(1 + 1)−Sk−1 for every k∈N. Using the proof of Lemma 2.6 we deduce that: 2(1 + 1)≥lim sup m lim sup n |||xn−ym||| ≥ 1 Sk lim sup n |||xn|||+1 S2 klim sup m |||ym−y||| +|||Qk(y)|||≥1 Sk +1 S2 k [1 + |||Qk(y)|||], which implies that |||Qk(y)||| ≤ 2S2 k(1 + 1)−Sk−1 for all y∈A∗(D). Choose x:= xνwith ν∈Nlarge enough so that x∈Dand |||x||| <1 + 1. Since the norm satisfies condition (1) in Definition 2.2, we know that limkSk= 1 and therefore limkµk= 21. Take k1∈Nso that |||Qk(x)||| < 1,and µk<31 if k≥k1. In particular, this implies that |||Qk1(y−x)||| ≤ |||Qk1(y)||| +|||Qk1(x)||| ≤ 31+1= 41<1, and |||Pk1(y−x)||| ≤ |||x−y||| +|||Qk1(y−x)||| ≤ |||x||| +|||y||| + 41 ≤1 + 1+ 4 + 41+ 41= 5 + 91< R0 for every y∈A∗(D). Given k1∈Nas before, there exists a corresponding function F(λ) := Fk1(λ) satisfying property (3) in Definition 2.2. Since limλ→0+F(λ) λ≤M R0, take λ∈(0,1) such that F(λ) λ<M+ 1 2R0 <1−41 R0 ≤c−41 R0 , which implies that (2 −λ)c+F(λ)R0+λ41<2c. Now, choose 2>0 with (2 −λ)(c+2) + F(λ)R0+λ41<2c. 8 CASTILLO-S´ ANTOS, DOWLING, FETTER, JAP ´ ON, LENNARD, SIMS, TURETT Choose (yn)⊂Dan a.f.p.s. with w∗-limnyn=yand such that lim sup n |||yn−y||| ≤ c+2. By passing to a subsequence, we may also suppose that lim inf n|||yn−y||| = lim sup n |||yn−y||| ≥ c≥1. Notice that, for every m∈N, the vectors (1 −λ)ym+λx ∈D. We claim that (∗∗) lim sup m lim sup n |||yn−[(1 −λ)ym+λx]||| <2c. Assume that (∗∗) holds. Then we can find some m∈Nsuch that lim sup n |||yn−[(1 −λ)ym+λx]||| <2c. This implies that for some r∈(0,2c) the set G:= {z∈D: lim sup n |||yn−z||| ≤ r} is a nonempty closed convex T-invariant subset of D, and therefore it contains an a.f.p.s. (zs), which tends to some z∈Xwith respect to the w∗-topology. In this case, using the definition of c, Lemma 2.6, and that each zs∈G, we have 2c≤lim sup s |||zs−z||| + lim sup n |||yn−y||| ≤ lim sup s lim sup n |||yn−zs||| ≤ r , which is a contradiction. We finish by proving the claim (∗∗). Noting that lim infn|||yn−y||| ≥ 1 and by property (3) in Definition 2.2, we have: lim supn|||(yn−y) + λPk1(y−x)||| ≤ lim supn|||yn−y||| +F(λ)|||Pk1(y−x)||| ≤c+2+F(λ)R0. Therefore, lim supmlim supn|||yn−[(1 −λ)ym+λx]||| = lim supmlim supn|||yn−y+y−(1 −λ)ym−λx||| = lim supmlim supn|||yn−y+ (1 −λ)y+λy −(1 −λ)ym−λx||| ≤lim supmlim supn[(1 −λ)|||ym−y||| +|||(yn−y) + λ(y−x)|||] ≤(1 −λ) lim supm|||ym−y||| + lim supn|||(yn−y) + λPk1(y−x)||| +λ|||Qk1(y−x)||| ≤(1 −λ)(c+2) + c+2+F(λ)R0+λ41 ≤(2 −λ)(c+2) + F(λ)R0+λ41 <2c , which proves (∗∗), and completes the proof of the theorem.  3. Norms with the Fixed Point Property Throughout this section, we will study several examples of norms which are near-infinity concentrated norms and therefore they satisfy the FPP according to Theorem 2.7. As a particular case of a more general result, we will deduce that (`1, νp(·)) has the FPP whenever νp(·) is a renorming of `1. We will start by proving that P.K. Lin’s norm is an example of a near-infinity concentrated norm. We will deduce this assertion from the following lemma. Lemma 3.1. Let (X, | · |)be a Banach space with a Schauder basis and assume that |·| satisfies properties (1) and (2) in Definition 2.1. If (γk)is a nondecreasing sequence in (0,1) converging to 1, then the norm defined as |x|1:= sup k γk|Qk(x)|,for all x∈X , NEAR-INFINITY CONCENTRATED NORMS AND THE FPP 9 is a near-infinity concentrated norm on Xthat is equivalent to |·|. Proof. Notice that γk|Qk(x)|≤|Qk(x)|1≤ |Qk(x)|for every k∈Nand x∈X, which implies that | · |1is a sequentially separating norm whenever | · | satisfies the same property. It is also easy to check that | · |1satisfies (2) in Definition 2.2. It remains to prove condition (3). Fix some k∈Nand R > 0. Let (xn) be a bounded pointwise-null sequence in Xwith lim infn|x|1≥1. Without loss of generality we can assume that Ql(xn) = Qk(xn) for every l≤k. Moreover, it is not difficult to check that lim supn|xn|= lim supn|xn|1. For every z∈Xwith Qk(z) = 0, |z|1≤Rand for every λ > 0 we have |xn+λz|1= suplγl|Ql(xn+λz)|= suplγl|Ql(xn) + λQl(z)| = max max 1≤l≤k−1γl|Ql(xn) + λQl(z)|,sup l≥k γl|Ql(xn)| = max max 1≤l≤k−1γl|Ql(xn) + λQl(z)|,|xn|1 ≤max{γk−1|xn|+λ|z|1,|xn|1} Taking limits when ngoes to infinity: lim sup n |xn+λz|1≤max{γk−1lim sup n |xn|1+λ|z|1,lim sup n |xn|1}. From above, lim supn|xn|1≥1 and |z|1≤R; and so lim sup n |xn+λz|1≤lim sup n |xn|1+Fk(λ)|z|1, where Fk(λ) := 0 if λ≤(1 −γk−1)/R, and Fk(λ) := λotherwise. Taking M= 0 and any R > 5 in Definition 2.2(3), we see that |·|1is a near infinity concentrated norm.  If we let |·|:= k · k1in `1we obtain that | · |1coincides with P.K. Lin’s norm ||| · |||L. Moreover, given a norm |·|0:= |·|satisfying (1) and (2) in Definition 2.2 and defining in a recursive way the equivalent norms |·|n= sup k γk|Qk(·)|n−1 for every n∈N, we can construct sequences of near-infinity concentrated norms. All of these norms |·|n(n≥1) satisfy the FPP when the basis is boundedly complete, according to Theorem 2.7. Lemma 3.2. Assume that (pn)⊂(1,+∞)is a nonincreasing sequence with limnpn= 1. Then the norm νp(·)is a near-infinity concentrated norm in the Banach space X, defined as the completion of c00 with the norm νp(·). Proof. Let us start by proving that νp(·) is a sequentially separating norm, that is, νp(·) satisfies property (1) in Definition 2.2. By Lemma 2.5, it suffices to check that limkSk(X, νp(·)) = 1. Fix k∈N. First note that if x= l X i=k x(i)eiwith k≤land l < y then νp(x+y) = νp l X i=k x(i)ei+νp(y)el+1!. On the other hand, it is not difficult to check that if 1 < p ≤qand a, b, c ≥0 then: k(a, k(b, c)kp)kq≥ k(a, k(b, c)kq)kq=aq+k(b, c)kq q1/q = (aq+bq+cq)1/q = =k(k(a, b)kq, c)kq.